 reserve n,m for Nat,
      o for object,
      p for pair object,
      x,y,z for Surreal;

theorem
  Union sqrtR(p,o)
    = R_p \/ sqrt(o,Union sqrtL(p,o),Union sqrtL(p,o)) \/
       sqrt(o,Union sqrtR(p,o),Union sqrtR(p,o))
proof
  defpred P[Nat] means
  sqrtR(p,o).$1 c= (R_p)\/sqrt(o,Union sqrtL(p,o),Union sqrtL(p,o))\/
       sqrt(o,Union sqrtR(p,o),Union sqrtR(p,o));
  sqrtR(p,o).0 = R_p by Th6;
  then
A1: P[0] by XBOOLE_1:7,10;
A2: P[n] implies P[n+1]
  proof
    assume
A3: P[n];
    let a be object;
    assume
A4: a in sqrtR(p,o).(n+1);
    sqrtR(p,o).(n+1) = sqrtR(p,o).n \/ sqrt(o,sqrtL(p,o).n,sqrtL(p,o).n)\/
      sqrt(o,sqrtR(p,o).n,sqrtR(p,o).n) by Th8;
    then a in sqrtR(p,o).n \/ sqrt(o,sqrtL(p,o).n,sqrtL(p,o).n) or a in
    sqrt(o,sqrtR(p,o).n,sqrtR(p,o).n) by XBOOLE_0:def 3,A4;
    then per cases by XBOOLE_0:def 3;
    suppose a in sqrtR(p,o).n;
      hence thesis by A3;
    end;
    suppose
A5:   a in sqrt(o,sqrtL(p,o).n,sqrtL(p,o).n);
      sqrtL(p,o).n c= Union sqrtL(p,o) by ABCMIZ_1:1;
      then sqrt(o,sqrtL(p,o).n,sqrtL(p,o).n) c=
      sqrt(o,Union sqrtL(p,o),Union sqrtL(p,o)) by Th11;
      then a in R_p\/sqrt(o,Union sqrtL(p,o),Union sqrtL(p,o))
      by A5,XBOOLE_0:def 3;
      hence thesis by XBOOLE_0:def 3;
    end;
    suppose
A6:   a in sqrt(o,sqrtR(p,o).n,sqrtR(p,o).n);
      sqrtR(p,o).n c= Union sqrtR(p,o) by ABCMIZ_1:1;
      then sqrt(o,sqrtR(p,o).n,sqrtR(p,o).n) c=
      sqrt(o,Union sqrtR(p,o),Union sqrtR(p,o)) by Th11;
      hence thesis by A6,XBOOLE_0:def 3;
    end;
  end;
A7: P[n] from NAT_1:sch 2(A1,A2);
  thus Union sqrtR(p,o)
    c= (R_p)\/sqrt(o,Union sqrtL(p,o),Union sqrtL(p,o))\/
       sqrt(o,Union sqrtR(p,o),Union sqrtR(p,o))
  proof
    let a be object;
    assume a in Union sqrtR(p,o);
    then consider n be object such that
A8: n in dom sqrtR(p,o) & a in sqrtR(p,o).n by CARD_5:2;
    n in NAT by A8,Def5;
    then reconsider n as Nat;
    sqrtR(p,o).n c= (R_p)\/sqrt(o,Union sqrtL(p,o),Union sqrtL(p,o))\/
       sqrt(o,Union sqrtR(p,o),Union sqrtR(p,o)) by A7;
    hence thesis by A8;
  end;
A9: sqrt(o,Union sqrtL(p,o),Union sqrtL(p,o)) c= Union sqrtR(p,o)
  proof
    let a be object;
    assume a in sqrt(o,Union sqrtL(p,o),Union sqrtL(p,o));
    then consider x1,y1 be Surreal such that
A10: x1 in Union sqrtL(p,o) & y1 in Union sqrtL(p,o) &
    not x1 + y1 == 0_No & a = (o +'(x1 * y1)) * (x1+y1)" by Def2;
    consider n be object such that
A11: n in dom sqrtL(p,o) & x1 in sqrtL(p,o).n by A10,CARD_5:2;
    consider m be object such that
A12: m in dom sqrtL(p,o) & y1 in sqrtL(p,o).m by A10,CARD_5:2;
    n in NAT & m in NAT by A12,A11,Def4;
    then reconsider n,m as Nat;
    set nm=n+m;
    m <= nm & n <= nm by NAT_1:11;
    then sqrtL(p,o).n c= sqrtL(p,o).nm & sqrtL(p,o).m c= sqrtL(p,o).nm by Th7;
    then a in sqrt(o,sqrtL(p,o).nm,sqrtL(p,o).nm) by A10,A11,A12,Def2;
    then
A13:a in sqrtR(p,o).nm \/ sqrt(o,sqrtL(p,o).nm,sqrtL(p,o).nm)
    by XBOOLE_0:def 3;
    sqrtR(p,o).(nm+1) = sqrtR(p,o).nm \/ sqrt(o,sqrtL(p,o).nm,sqrtL(p,o).nm)
    \/ sqrt(o,sqrtR(p,o).nm,sqrtR(p,o).nm) by Th8;
    then
A14:a in sqrtR(p,o).(nm+1) by A13,XBOOLE_0:def 3;
    nm+1 in NAT;
    then nm+1 in dom sqrtR(p,o) by Def5;
    hence thesis by A14,CARD_5:2;
  end;
A15: sqrt(o,Union sqrtR(p,o),Union sqrtR(p,o)) c= Union sqrtR(p,o)
  proof
    let a be object;
    assume a in sqrt(o,Union sqrtR(p,o),Union sqrtR(p,o));
    then consider x1,y1 be Surreal such that
A16: x1 in Union sqrtR(p,o) & y1 in Union sqrtR(p,o) &
    not x1 + y1 == 0_No & a = (o +'(x1 * y1)) * (x1+y1)" by Def2;
    consider n be object such that
A17: n in dom sqrtR(p,o) & x1 in sqrtR(p,o).n by A16,CARD_5:2;
    consider m be object such that
A18: m in dom sqrtR(p,o) & y1 in sqrtR(p,o).m by A16,CARD_5:2;
    n in NAT & m in NAT by A18,A17,Def5;
    then reconsider n,m as Nat;
    set nm=n+m;
    m <= nm & n <= nm by NAT_1:11;
    then sqrtR(p,o).n c= sqrtR(p,o).nm & sqrtR(p,o).m c= sqrtR(p,o).nm by Th7;
    then
A19:a in sqrt(o,sqrtR(p,o).nm,sqrtR(p,o).nm) by A16,A17,A18,Def2;
    sqrtR(p,o).(nm+1) = sqrtR(p,o).nm \/ sqrt(o,sqrtL(p,o).nm,sqrtL(p,o).nm)
    \/ sqrt(o,sqrtR(p,o).nm,sqrtR(p,o).nm) by Th8;
    then
A20:a in sqrtR(p,o).(nm+1) by A19,XBOOLE_0:def 3;
    nm+1 in NAT;
    then nm+1 in dom sqrtR(p,o) by Def5;
    hence thesis by A20,CARD_5:2;
  end;
  R_p = sqrtR(p,o).0 by Th6;
  then R_p c= Union sqrtR(p,o) by ABCMIZ_1:1;
  then (R_p)\/sqrt(o,Union sqrtL(p,o),Union sqrtL(p,o)) c= Union sqrtR(p,o)
  by A9,XBOOLE_1:8;
  hence thesis by XBOOLE_1:8,A15;
end;
